Solution of y''-(1+x)y=0 using power series ..

Hello

Obtain the solution of the following differential equation as a power series about the origin :

$\displaystyle y''-(1+x)y=0$

Solution:

Let $\displaystyle \displaystyle y=\sum_{n=0}^{\infty} a_n x^n$

then :

$\displaystyle \displaystyle y'=\sum_{n=1}^{\infty} n a_n x^{n-1}$

$\displaystyle \displaystyle y''=\sum_{n=2}^{\infty} n (n-1) a_n x^{n-2}$

Rewrite the equation as : $\displaystyle y''-y -xy=0$

and substitute the values of y & y'' gives:

$\displaystyle \displaystyle \sum_{n=2}^{\infty} n (n-1) a_n x^{n-2} - \sum_{n=0}^{\infty} a_n x^n - \sum_{n=0}^{\infty} a_n x^{n+1} $

or:

$\displaystyle \displaystyle \sum_{n=0}^{\infty} (n+2)(n+1) a_{n+2} x^n - \sum_{n=0}^{\infty} a_n x^n - \sum_{n=0}^{\infty} a_n x^{n+1} $

My problem is in the last summation

I want to make the summations start with the same n and the powers of the x's are n in order to get the recursive relation, but I can not do that .

the last summation makes my life hard

any help?